In this case, the states of the model are the three possible types of weather: sunny, rainy and foggy. We assume the weather can be only one of these values. Thus the set of HMM states are:
S = {sunny, rainy, foggy}
Here, we can't observe the weather directly. Instead, the only evidence we have is whether the person every day is carrying an umbrella or not. In HMM terminology, these are the discrete observations:
x = {umbrella, no umbrella}
The HMM model is determined by three things:
The prior probabilities: vector of probabilities of being in the first state of a sequence.
The transition prob: matrix describing the probabilities of going from one state of the weather to another.
The emission prob: matrix describing the probabilities of observing an output (umbrella or not) given a state (weather).
We may be given with these probabilities, or we have to learn them from a training set. Then, perform reasoning like computing likelihood of an observation sequence with respect to an HMM model
Some steps to follow:
1) known model parameters
Check out this sample code that shows how to fill existing probabilities to build the model:
Q = 3; # number of states (sun,rain,fog)
O = 2; # number of discrete observations (umbrella, no umbrella)
# prior probabilities
prior = [1 0 0];
# state transition matrix (1: sun, 2: rain, 3:fog)
A = [0.8 0.05 0.15; 0.2 0.6 0.2; 0.2 0.3 0.5];
# observation emission matrix (1: umbrella, 2: no umbrella)
B = [0.1 0.9; 0.8 0.2; 0.3 0.7];
Then we can sample a bunch of sequences from this model:
num = 20; # 20 sequences
T = 10; # each of length 10 (days)
[seqs,states] = dhmm_sample(prior, A, B, num, T);
For example
>> seqs(5,:) # observation sequence
ans =
2 2 1 2 1 1 1 2 2 2
>> states(5,:) # hidden states sequence
ans =
1 1 1 3 2 2 2 1 1 1
Then, evaluate the log-likelihood of the sequence:
dhmm_logprob(seqs(5,:), prior, A, B)
dhmm_logprob_path(prior, A, B, states(5,:))
Compute the Viterbi path (most probable state sequence):
vPath = viterbi_path(prior, A, multinomial_prob(seqs(5,:),B))
2) unknown model parameters
Training is performed using the EM algorithm and is best done with a set of observation sequences.
We can use the generated data above to train a new model and compare it to the original:
# we start with a randomly initialized model
prior_hat = normalise(rand(Q,1));
A_hat = mk_stochastic(rand(Q,Q));
B_hat = mk_stochastic(rand(Q,O));
# learn from data by performing many iterations of EM
[LL,prior_hat,A_hat,B_hat] = dhmm_em(seqs, prior_hat,A_hat,B_hat, 'max_iter',50);
# plot learning curve
plot(LL), xlabel('iterations'), ylabel('log likelihood'), grid on
In this example, the trained model looks close to the original one:
>> p = [2 3 1]; # states permutation
>> prior, prior_hat(p)
prior =
1 0 0
ans =
0.97401
7.5499e-005
0.02591
>> A, A_hat(p,p)
A =
0.8 0.05 0.15
0.2 0.6 0.2
0.2 0.3 0.5
ans =
0.75967 0.05898 0.18135
0.037482 0.77118 0.19134
0.22003 0.53381 0.24616
>> B, B_hat(p,[1 2])
B =
0.1 0.9
0.8 0.2
0.3 0.7
ans =
0.11237 0.88763
0.72839 0.27161
0.25889 0.74111
To know more about Hidden Markov Model, study Artificial Intelligence Course.
Hope this answer helps you!
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